Adaptive predefined-time precise anti-unwinding attitude control for a spacecraft with inertia uncertainty

Abstract

This paper proposes an adaptive predefined-time accurate anti-unwinding attitude controller for a spacecraft with uncertain inertia. The novelty of the proposed algorithm lies in that the spacecraft attitude error is regulated to exact zero after a predefined settling time, which is an explicit parameter in the control algorithm. A time mapping function containing the predefined settling time is introduced, thus transforming the time scale of the original attitude control system. Then a composite barrier Lyapunov function is constructed to avoid the unwinding phenomenon in the attitude control process, and an adaptive backstepping attitude controller is designed based on the attitude system in the new time scale. The stability analysis of the attitude system is carried out based on LaSalle-Yoshizawa theorem, and it is proved that the attitude quaternion and angular rate can converge to zero accurately within the desired predefined time without using the spacecraft inertia information. Numerical simulation results show the effectiveness and superiority of the proposed controller.

Introduction

With the current rapid advancement of space science and technology, spacecrafts, such as artificial satellites and space stations, are critical tools for space exploration. Meanwhile, the number of space debris has risen considerably as a result of satellite failure, the shedding of launch vehicle bodies, and the explosion and collision of numerous types of spacecrafts. According to the data in Refs.,^1,2 the earth orbit is populated with over 36,500 orbital debris objects exceeding 10 cm, around a million fragments in 1–10 cm range. As a result, it is critical for some spacecrafts for capture to trap space debris and transport it into the atmosphere for destruction, freeing up orbital resources and ensuring the flight safety of other spacecrafts. Therefore, spacecraft capture and control technology has been one of the research hotspots. ³ The captured space debris are usually non-cooperative targets that have been affected by solar light pressure and gravity gradient for a long time, resulting in a complicated rotational motion, ⁴ which may cause the rotation of the integrated spacecraft system after a capture mission. To ensure the long-term on-orbit service of the spacecrafts for capture, their attitude must be stabilized rapidly for the next mission. ⁵ In this regard, researchers have developed numerous control approaches, including sliding-mode control, ⁶ adaptive control,^7,8 robust control, ⁹ and prescribed performance control.^10,11

Because the inertia information of the captured non-cooperative targets is usually unknown, the general inertia of the integrated spacecraft system after capture is uncertain. Refs.^12–14 regarded the inertia uncertainties of the integrated spacecraft system after capture as bounded perturbations, and they developed attitude control techniques using the known basis inertia. However, for the spacecrafts after executing capture missions frequently, the discrepancy between their true inertia and the known basis inertia may be substantial, so it may be difficult for the control algorithms based on the basis inertia information to fulfill the attitude accuracy requirements. ¹⁵ As a result, if the spacecraft attitude control issue with uncertain inertia is considered, adaptive control technique is always employed.^16–18 Lin and Zhang ¹⁶ proposed a robust adaptive attitude controller, using neural networks to approximate the inertia uncertainty. Shi et al. ¹⁷ developed an adaptive sliding mode attitude control technique for a spacecraft with uncertain inertia. Zhang and Li ¹⁸ proposed an adaptive law to estimate the unknown inertia parameter in a spacecraft formation system.

The control efficiency is another problem that should be considered in the spacecraft attitude control system after capturing non-cooperative targets. Zhao et al. ¹⁹ and Ye et al. ²⁰ proposed finite-time nonsingular sliding mode attitude control algorithms. Gao et al. ²¹ and Sun et al. ²² proposed fixed-time fast attitude control algorithms for spacecraft systems, considering actuator faults and quantized input signal, respectively. Although the finite-time and fixed-time control methods can improve the attitude control efficiency, the settling times of spacecraft attitude control systems under the traditional finite-time control methods^19,20 were related to the initial attitude errors, and the settling times of spacecraft attitude control systems under the fixed-time control methods^21,22 were affected by several control parameters. Thus, it may be difficult to know and set the upper bound of settling time directly by using the traditional finite-time and fixed-time spacecraft attitude control methods.^19–22

In recent years, researchers have focused on a new class of systems called predefined-time stable systems.^23–27 The upper bound of settling time of a predefined-time stable system is an explicit parameter in the control algorithm, thus allowing users to know and set its value in directly and quantitatively. Because of this considerable advantage, predefined-time control has been applied to the spacecraft attitude control issue.^28–35 Wang et al. ²⁸ and Xie et al. ²⁹ proposed predefined-time attitude controllers based on predefined-time Lyapunov dynamics. Shi et al. ³⁰ and Su and Shen ³¹ applied prescribed performance control approach to limit the settling time and the steady error of attitude control systems into the desired ranges. Ye et al. ³² guaranteed that the spacecraft attitude tracking error converges to a region with predefined boundary within a predetermined settling time. When the inertia uncertainty is considered in the predefined-time attitude control issue, Sun et al. ³³ designed a predefined-time attitude controller using the basis inertia information. Moreover, Xie et al.^34,35 applied neural networks and Fuzzy logic systems to approximate inertia uncertainties, regulating attitude errors to the neighborhood of zero within predefined settling times. Although the issue of predefined-time attitude control was studied in several previous studies, it can be seen that it is difficult for the control methods in Refs.^30–35 to achieve zero attitude error after the predefined settling times, so the control accuracy needs to be improved. Further, Refs.^{28,29,31–33} assumed that the spacecraft inertia information or its basis value is known, so they may not be applicable to the spacecrafts with uncertain inertia after capturing non-cooperative targets frequently. Therefore, a novel adaptive attitude controller without using the unknown inertia information should be investigated to eliminate the attitude error after the predefined settling time theoretically.

Considering the three difficulties, that is, the inertia uncertainty, the direct settling time adjustment, and the accurate attitude control simultaneously, we propose a new adaptive precise predefined-time attitude controller for a spacecraft in this study. It is required that the upper bound of settling time is an explicit control parameter, and that the attitude error after the predefined settling time is exact zero. A time mapping function is introduced to transform the time scale of the original attitude control system. The adaptive anti-unwinding backstepping attitude control algorithm is designed by using a composite barrier Lyapunov function. The system stability is proven by applying LaSalle-Yoshizawa theorem. Finally, numerical simulations are performed to validate the effectiveness and superiority of the presented method. The main contributions of the paper can be summarized as follows:

• (1) The predefined settling time of the spacecraft attitude system is an explicit parameter in the control algorithm, so the users can know and set the desired upper bound of settling time directly with the proposed method, comparing it with traditional finite-time and fixed-time methods.^19–22

• (2) The proposed controller guarantees zero attitude error after the predefined settling time theoretically, thus improving the control accuracy, compared with traditional predefined-time attitude controllers.^30–35

• (3) Compared with traditional predefined-time attitude controllers,^{28,29,31–33} the proposed controller does not use the unknown inertia information or its basis value of the spacecraft. It can be applied to controlling the spacecrafts after capturing non-cooperative targets.

The rest of the paper is organized as follows: Section 2 presents some preliminaries and describes the control objectives of the paper. Section 3 specifies the detailed adaptive precise predefined-time attitude control algorithm. Section 4 verifies the proposed method with numerical simulations and Section 5 concludes the paper.

Preliminaries and control objectives

Preliminaries

The relationship between the inertial coordinate frame $(X_I,Y_I,Z_I)$ and the spacecraft body coordinate frame $(X_B,Y_B,Z_B)$ can be depicted in Figure 1. The mathematical model of a spacecraft attitude control system can be established as^7,8,11:

$$\begin{array}{c}\multicolumn{1}{c}{\stackrel{·}{q}=\frac{1}{2}(q^{\times }+q_0{I}_3)\omega ,{\stackrel{·}{q}}_0=-\frac{1}{2}{\omega }^{T}q}\\ \multicolumn{1}{c}{J\stackrel{·}{\omega }=-{\omega }^{\times }J\omega +u}\end{array}$$

(1)

where $q_0\in \mathbb{R}$ and $q={[q_1,q_2,q_3]}^{\mathrm{T}}\in {\mathbb{R}}^3$ are the attitude quaternion of the spacecraft body coordinate system $\mathcal{B}$ with respect to the inertia coordinate system $\mathcal{I}$ , and they fulfill the equation constraint $q^{\mathrm{T}}q+q_0^2=1$ ; $\omega ={[{\omega }_1,{\omega }_2,{\omega }_3]}^{\mathrm{T}}\in {\mathbb{R}}^3$ represents the angular velocity of the spacecraft with respect to the inertia coordinate system $\mathcal{I}$ ; $J\in {\mathbb{R}}^{3\times 3}$ denotes the unknown inertia matrix of the spacecraft, with

$$J=\left[\begin{array}{ccc}{J}_{11}\hfill & J_{12}\hfill & J_{13}\hfill \\ \multicolumn{1}{c}{J_{12}}& J_{22}\hfill & J_{23}\hfill \\ \multicolumn{1}{c}{J_{13}}& J_{23}\hfill & J_{33}\hfill \end{array}\right]$$

(2)

where $J_{\mathrm{ik}}\in \mathbb{R}$ are unknown constant parameters ( $i=1,2,3;k=1,2,3$ ); $u={[u_1,u_2,u_3]}^{\mathrm{T}}\in {\mathbb{R}}^3$ denotes the control torques; $I_3\in {\mathbb{R}}^{3\times 3}$ denotes the unit matrix; besides, for any vector $a={[a_1,a_2,a_3]}^{\mathrm{T}}\in {\mathbb{R}}^3$ , the cross operator for $a$ , that is, $a^{\times }\in {\mathbb{R}}^{3\times 3}$ , is defined as:

$$a^{\times }=\left[\begin{array}{ccc}0\hfill & -a_3\hfill & a_2\hfill \\ \multicolumn{1}{c}{a_3}& 0\hfill & -a_1\hfill \\ \multicolumn{1}{c}{-a_2}& a_1\hfill & 0\hfill \end{array}\right].$$

(3)

Figure 1.

The relationship between the inertial coordinate frame and the spacecraft body coordinate frame.

Then, the following assumption and two lemmas are provided for the control design and the stability analysis of the spacecraft attitude system.

• Assumption 1.^28,29 The real-time information of the spacecraft attitude quaternion $q_0(t)$ , $q(t)$ and the angular velocity $\omega (t)$ is available for feedback control design.

• Lemma 1. (LaSalle-Yoshizawa theorem ³⁶ ) Consider a nonlinear system $\stackrel{·}{\xi }\left(\lambda \right)=f(\xi ,\lambda )$ , where $f:{\mathbb{R}}^n\times [0,+\infty )\to {\mathbb{R}}^n$ is locally Lipschitz in $\xi$ , uniformly in $\lambda$ . Let $\xi =0$ be the equilibrium point of $\stackrel{·}{\xi }\left(\lambda \right)=f(\xi ,\lambda )$ . Let $L(\xi ,\lambda ):{\mathbb{R}}^n\times [0,+\infty )\to [0,+\infty )$ be a continuously differentiable function, such that, for all $\lambda \in [0,+\infty )$ , there are

$${\gamma }_1(\Vert \xi {\Vert }_2)\le L(\xi ,\lambda )\le {\gamma }_2(\Vert \xi {\Vert }_2)$$

(4)

$$\frac{\mathrm{d}L(\xi ,\lambda )}{\mathrm{d}\lambda }=\frac{\partial L(\xi ,\lambda )}{\partial \xi }\frac{\mathrm{d}\xi }{\mathrm{d}\lambda }+\frac{\partial L(\xi ,\lambda )}{\partial \lambda }\le -W\left(\xi \right)\le 0$$

(5)

where ${\gamma }_1$ and ${\gamma }_2$ are class ${\mathcal{K}}_{\infty }$ functions, and $W$ is a continuous function with respect to $\xi$ , then the solutions of $\stackrel{·}{\xi }\left(\lambda \right)=f(\xi ,\lambda )$ are globally uniformly bounded and satisfy

$$\underset{\lambda \to \infty }{\lim }W\left(\xi \left(\lambda \right)\right)=0.$$

(6)

• Lemma 2. (Predefined-time stability^24,26) Consider a nonlinear non-autonomous system $\stackrel{·}{y}\left(t\right)=g(y\left(t\right),t,T)$ with initial condition $y\left(0\right)=y_0\in {\mathbb{R}}^n$ , where $y\left(t\right)\in {\mathbb{R}}^n$ is the system state vector, $T>0$ denotes a constant parameter, and $g:{\mathbb{R}}^n\times [0,+\infty )\times (0,+\infty )\to {\mathbb{R}}^n$ is a nonlinear function. Denote the solutions of the system as $y(y\left(0\right),t,T)$ . Then the system $\stackrel{·}{y}\left(t\right)=g(y\left(t\right),t,T)$ is globally predefined-time stable if for any $y_0\in {\mathbb{R}}^n$ , $y(y\left(0\right),t,T)=0$ holds for all $t\ge T$ . Besides, the constant parameter $T$ is the predefined settling time.

• Remark 1. Consider a controlled system $\stackrel{·}{y}\left(t\right)=g(y\left(t\right),v(t,T))$ with initial condition $y\left(0\right)=y_0\in {\mathbb{R}}^n$ , where $v(t,T)=v(y\left(t\right),t,T)$ is a feedback control law, and $T>0$ is a constant parameter involved in the control law. If the system $\stackrel{·}{y}\left(t\right)=g(y\left(t\right),v(t,T))$ is predefined-time stable, that is, for any $y_0\in {\mathbb{R}}^n$ , $y\left(t\right)=0$ holds for all $t\ge T$ , then the predefined settling time of the system is the explicit parameter $T$ in the control law, so its value can be known and set directly by users.

Control objective

The control objectives of this paper is to design an adaptive feedback controller $u$ for the spacecraft attitude control system (1), such that the predefined-time accurate anti-unwinding attitude control can be achieved under unknown spacecraft inertia information. The control objective can be detailed as:

• (i) System (1) is predefined-time stable within a predefined settling time $T$ , that is,

$$\{\begin{array}{c}q\left(t\right)=0\\ \underset{t\to }{\lim }\mathrm{T}\omega \left(t\right)=0,\end{array}t\ge T$$

(7)

where $T>0$ is an explicit parameter that appears in the control algorithm directly;

• (ii) Attitude unwinding phenomenon can be avoided in the control process, that is,

$$\{\begin{array}{c}\underset{t\to T}{\lim }{q}_0\left(t\right)=\mathrm{sign}\left(q_0\left(0\right)\right),q_0\left(0\right)\ne 0\hfill \\ \multicolumn{1}{c}{\underset{t\to T}{\lim }{q}_0\left(t\right)=\pm 1,q_0\left(0\right)=0;}\end{array}$$

(8)

• (iii) The control algorithm does not utilize the unknown spacecraft inertia information $J$ .

• Remark 2. The problem of spacecraft attitude unwinding refers to the phenomenon when the initial attitude of the spacecraft is close to the expected attitude, the improper control law may cause the spacecraft to rotate away from the expected attitude and then return to it. ³⁷ When the initial attitude is close to the intended attitude, the phenomenon may induce excessive rotation in the entire attitude maneuver. As a result, the phenomenon of unwinding might cause excessive fuel consumption.

Control design

This section presents the adaptive predefined-time accurate anti-unwinding attitude controller and the system stability analysis for system (1). The structure of the control scheme is shown in Figure 2.

Figure 2.

The structure of the proposed control scheme.

Model transformation

In order to stabilize the spacecraft attitude system (1) within the predefined settling time $T$ , a time mapping function is introduced, transforming the time scale of system (1) before the predefined settling time $T$ .

The time mapping function is established as

$$s\left(t\right)=-T\ln \frac{T-t}{T},t<T.$$

(9)

The inverse mapping of the time mapping function (9) can be given by

$$t\left(s\right)=T-T{e}^{-\frac{s}{T}},s\in [0,+\infty ].$$

(10)

The relationship between the original time scale $t$ and the transformed time scale $s$ is shown in Figure 3. It can be seen from Figure 3, (9) and (10) that the time mapping function $s\left(t\right)$ and its inverse function $t\left(s\right)$ satisfy the following properties:

$$\{\begin{array}{c}\underset{t\to 0}{\lim }s\left(t\right)=0,{\lim }_{s\to 0}t\left(s\right)=0\hfill \\ \multicolumn{1}{c}{\underset{t\to T}{\lim }s\left(t\right)\to +\infty ,\underset{s\to +\infty }{\lim }t\left(s\right)=T.}\end{array}$$

(11)

• Remark 3. Due to the uniformity between the original time period $t\in [0,T)$ and the transformed time period $s\in [0,+\infty )$ . The predefined-time control objective with respect to the original time scale $t$ can be converted to the asymptotic control objective with respect to the transformed time scale $s$ , that is, if $\mathrm{li}{\mathrm{m}}_{s\to +\infty }q\left(s\right)=\mathrm{li}{\mathrm{m}}_{s\to +\infty }\omega \left(s\right)=0$ , then we have $\mathrm{li}{\mathrm{m}}_{t\to T}q\left(t\right)=\mathrm{li}{\mathrm{m}}_{t\to T}\omega \left(t\right)=0$ .

Figure 3.

Curves of time mapping (9) and its inverse mapping (10): (a) curve of time mapping and (b) curve of inverse mapping.

Considering the transformed time $s$ as the new time scale of system (1) in $t<T$ , we can rewrite system (1) according to (10), as

$$\begin{array}{c}\frac{\mathrm{d}q}{\mathrm{d}s}=\frac{\mathrm{d}q}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}s}=\frac{1}{2}{\mathrm{e}}^{-\frac{s}{T}}(q^{\times }+q_0{I}_3)\omega ,\hfill \\ \frac{\mathrm{d}{q}_0}{\mathrm{d}s}=\frac{\mathrm{d}{q}_0}{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}s}=-\frac{1}{2}{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{\mathrm{T}}q\hfill \\ J\frac{\mathrm{d}\omega }{\mathrm{d}s}=J\frac{\mathrm{d}\omega }{\mathrm{d}t}\frac{\mathrm{d}t}{\mathrm{d}s}=-{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{\times }J\omega +{\mathrm{e}}^{-\frac{s}{T}}u.\hfill \end{array}$$

(12)

Then, for any vector $a={[a_1,a_2,a_3]}^{\mathrm{T}}\in {\mathbb{R}}^3$ , we define a matrix $L\left(a\right)\in {\mathbb{R}}^{3\times 6}$ as

$$L\left(a\right)=\left[\begin{array}{cccccc}{a}_1\hfill & 0\hfill & 0\hfill & 0\hfill & a_3\hfill & a_2\hfill \\ \multicolumn{1}{c}{0}& a_2\hfill & 0\hfill & a_3\hfill & 0\hfill & a_1\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & a_3\hfill & a_2\hfill & a_1\hfill & 0\hfill \end{array}\right].$$

(13)

For the sake of the estimation of unknown inertia matrix $J$ , an unknown constant vector $J^{*}\in {\mathbb{R}}^6$ is defined as

$$J^{*}={[J_{11},J_{22},J_{33},J_{23},J_{13},J_{12}]}^{\mathrm{T}}.$$

(14)

To proceed, the term $J\omega$ in (12) satisfies

$$J\omega =\left[\begin{array}{c}{J}_{11}{\omega }_1+J_{12}{\omega }_2+J_{13}{\omega }_3\hfill \\ \multicolumn{1}{c}{J_{12}{\omega }_1+J_{22}{\omega }_2+J_{23}{\omega }_3}\\ \multicolumn{1}{c}{J_{13}{\omega }_1+J_{23}{\omega }_2+J_{33}{\omega }_3}\end{array}\right]=L\left(\omega \right)J^{*}.$$

(15)

Thus, system (12) in $t<T$ is converted to be

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}q}{\mathrm{d}s}=\frac{1}{2}{\mathrm{e}}^{-\frac{s}{T}}(q^{\times }+q_0{I}_3)\omega ,\frac{\mathrm{d}{q}_0}{\mathrm{d}s}=-\frac{1}{2}{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{T}q}\\ \multicolumn{1}{c}{J\frac{\mathrm{d}\omega }{\mathrm{d}s}=-{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{\times }L\left(\omega \right)J^{*}+{\mathrm{e}}^{-\frac{s}{T}}u.}\end{array}$$

(16)

In this paper, the control law $u$ in $t<T$ will be designed based on the transformed system (16) with respect to the transformed time scale $s$ for predefined-time control.

Adaptive accurate predefined-time attitude control algorithm design

Adaptive backstepping control is applied to design the control law $u$ for system (16) in $t<T$ . Define the angular velocity tracking error $z\in {\mathbb{R}}^3$ and the estimation error $\tilde{J}\in {\mathbb{R}}^6$ for unknown constant vector $J^{*}$ , respectively, as

$$\begin{array}{c}z=\omega -\alpha \hfill \\ \multicolumn{1}{c}{\tilde{J}=J^{*}-\hat{J}}\end{array}$$

(17)

where $\alpha \in {\mathbb{R}}^3$ is a virtual control law to be designed, and $\hat{J}\in {\mathbb{R}}^6$ is the estimation of unknown constant vector $J^{*}$ . Then the control law design process is carried out with the following two steps:

• Step 1. In order to achieve the anti-unwinding attitude control for system (16) and to consider both zero and non-zero initial conditions of $q_0\left(t\right)$ , the following composite barrier Lyapunov function is constructed:

$$V_1=\{\begin{array}{c}-k\ln {q_0}^2+q^{\mathrm{T}}q,q_0\left(0\right)\ne 0\hfill \\ \multicolumn{1}{c}{{(q_0-1)}^2+q^{\mathrm{T}}q,q_0\left(0\right)=0}\end{array}$$

(18)

where $k>0$ is a design parameter. The curves of the composite barrier Lyapunov function (18) is descripted in Figure 4. It can be seen from Figure 4 that $\underset{q^{\mathrm{T}}q\to 1}{lim}{V}_1\to +\infty$ in the case $q(0)\ne 0$ , and that $V_1$ is a quadratic Lyapunov function with respect to $q$ in the case $q(0)=0$ .

Figure 4.

Curves of the composite barrier Lyapunov function (18): (a) in case q₀ (0) ≠ 0 and (b) in case q₀ (0) = 0.

Differentiating $V_1$ with respect to the transformed time scale $s$ and substituting (16) and (17) into it yield

$$\begin{array}{c}\frac{\mathrm{d}{V}_1}{\mathrm{d}s}={\mathrm{e}}^{-\frac{s}{T}}K\left(q_0\right)q^{\mathrm{T}}\omega ={\mathrm{e}}^{-\frac{s}{T}}K\left(q_0\right)q^{\mathrm{T}}(z+\alpha )\hfill \end{array}$$

(19)

where $K\left(q_0\right)\in \mathbb{R}$ is a variable determined by the initial value of $q_0\left(t\right)$ , and it is given by

$$K\left(q_0\right)=\{\begin{array}{cc}{q}_0+k{q}_0^{-1},& q_0\left(0\right)\ne 0\\ 1& q_0\left(0\right)=0.\end{array}$$

(20)

The virtual control law $\alpha$ is designed as

$$\begin{array}{c}\alpha =-c_1{K}^{-1}\left(q_0\right){\mathrm{e}}^{\frac{s}{T}}q\hfill \end{array}$$

(21)

where $c_1>0$ is a design parameter.

Substituting (21) into (19) leads to

$$\begin{array}{c}\frac{\mathrm{d}{V}_1}{\mathrm{d}s}=-c_1{q}^{\mathrm{T}}q+{\mathrm{e}}^{-\frac{s}{T}}K\left(q_0\right)q^{\mathrm{T}}z.\hfill \end{array}$$

(22)

• Step 2. The closed-loop system Lyapunov function for (16) can be established by using (17) and (18) as

$$V=V_1+\frac{1}{2}{z}^{\mathrm{T}}\mathrm{Jz}+\frac{1}{2\gamma }{\tilde{J}}^{\mathrm{T}}\tilde{J}$$

(23)

where $\gamma >0$ is a design parameter.

The derivative of $V$ with respect to the transformed time scale $s$ can be obtained from (16), (17), and (22), as

$$\begin{array}{c}\multicolumn{1}{c}{\frac{\mathrm{d}V}{\mathrm{d}s}=-c_1{q}^{\mathrm{T}}q+{\mathrm{e}}^{-\frac{s}{T}}K(q_0)q^{\mathrm{T}}z+z^{\mathrm{T}}(J\frac{\mathrm{d}\omega }{\mathrm{d}s}-J\frac{\mathrm{d}\alpha }{\mathrm{d}s})-\frac{1}{\gamma }{\tilde{J}}^{\mathrm{T}}\frac{\mathrm{d}\hat{J}}{\mathrm{d}s}}\\ =-c_1{q}^{\mathrm{T}}q+{\mathrm{e}}^{-\frac{s}{T}}K(q_0)q^{\mathrm{T}}z+z^{\mathrm{T}}\hfill \\ [-{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{\times }L(\omega )J^{*}+{\mathrm{e}}^{-\frac{s}{T}}u-L\left(\frac{\mathrm{d}\alpha }{\mathrm{d}s}\right)J^{*}]-\frac{1}{\gamma }{\hat{J}}^{\mathrm{T}}\frac{\mathrm{d}\hat{J}}{\mathrm{d}s}\hfill \\ =-c_1{q}^{\mathrm{T}}q+{\mathrm{e}}^{-\frac{s}{T}}K(q_0)q^{\mathrm{T}}z+z^{\mathrm{T}}\hfill \\ [-{\mathrm{e}}^{-\frac{s}{T}}{\omega }^{\times }L(\omega )\hat{J}+{\mathrm{e}}^{-\frac{s}{T}}u-L\left(\frac{\mathrm{d}\alpha }{\mathrm{d}s}\right)\hat{J}]\hfill \\ -\frac{1}{\gamma }{\tilde{J}}^{\mathrm{T}}[\frac{\mathrm{d}\hat{J}}{\mathrm{d}s}+\gamma {\mathrm{e}}^{-\frac{s}{T}}{L}^{\mathrm{T}}(\omega ){\omega }^{\times \mathrm{T}}z+\gamma L^{\mathrm{T}}\left(\frac{\mathrm{d}\alpha }{\mathrm{d}s}\right)z]\hfill \end{array}$$

(24)

where $J\mathrm{d}\alpha /\mathrm{d}s=L(\mathrm{d}\alpha /\mathrm{d}s)J^{*}$ is applied according to (15), and $\mathrm{d}\alpha /\mathrm{d}s$ can be obtained from (16), (20), and (21), as

$$\begin{array}{c}\frac{\mathrm{d}\alpha }{\mathrm{d}s}=-c_1{\mathrm{e}}^{\frac{s}{T}}Y\left(q_0\right)\frac{\mathrm{d}{q}_0}{\mathrm{d}s}q-c_1{\mathrm{e}}^{\frac{s}{T}}{K}^{-1}\left(q_0\right)\frac{\mathrm{d}q}{\mathrm{d}s}\hfill \\ -c_1{\mathrm{e}}^{\frac{s}{T}}{T}^{-1}{K}^{-1}\left(q_0\right)q\hfill \end{array}$$

(25)

with

$$\begin{array}{c}\multicolumn{1}{c}{Y\left(q_0\right)=\{\begin{array}{cc}\frac{k-q_0^2}{{(k+q_0^2)}^2},\hfill & q_0\left(0\right)\ne 0\hfill \\ \multicolumn{1}{c}{0,}& q_0\left(0\right)=0.\hfill \end{array}}\end{array}$$

(26)

Design the control law $u$ and the adaptive law $\mathrm{d}\hat{J}/\mathrm{d}s$ , respectively, as

$$u={\mathrm{e}}^{\frac{s}{T}}[-c_{2}z+L\left(\frac{\mathrm{d}\alpha }{\mathrm{d}s}\right)\hat{J}]-K\left(q_0\right)q+{\omega }^{\times }L\left(\omega \right)\hat{J}$$

(27)

$$\frac{\mathrm{d}\hat{J}}{\mathrm{d}s}=-\gamma {\mathrm{e}}^{-\frac{s}{T}}{L}^T\left(\omega \right){\omega }^{\times T}z-\gamma L^T\left(\frac{\mathrm{d}\alpha }{\mathrm{d}s}\right)z,\hat{J}\left(0\right)=0.$$

(28)

Substituting (27) and (28) into (24) yields

$$\frac{\mathrm{d}V}{\mathrm{d}s}=-c_1{q}^{T}q-c_2{z}^{T}z\le 0.$$

(29)

This ends the adaptive accurate predefined-time anti-unwinding backstepping attitude control law design in $t<T$ .

In $t\ge T$ , the following control law is to be applied to maintain system (1) satisfying $q\left(t\right)=\omega \left(t\right)=0$ :

$$\begin{array}{c}u\left(t\right)=0,t\ge T.\hfill \end{array}$$

(30)

System analysis

In this subsection, a theorem is introduced to analyze the predefined-time stable property and the anti-unwinding characteristic of the spacecraft attitude system (1).

• Theorem 1. Consider the spacecraft attitude control system (1) and its transformed system (16) the virtual control law (21), the control laws (27) and (30), and the adaptive law (28). The control objectives (7) and (8) hold.

• Proof. The proof of this theorem is carried out in four steps as follows:

• Step 1. To prove the predefined-time stability of $q\left(t\right)$ and $z\left(t\right)$ based on LaSalle-Yoshizawa theorem.

Denote $x\left(s\right)=[q^{\mathrm{T}}\left(s\right),z^{\mathrm{T}}\left(s\right),{\tilde{J}}^{\mathrm{T}}\left(s\right){]}^{\mathrm{T}}\in {\mathbb{R}}^{12}$ as the closed-loop system error vector of system (16). According to (17), (18), (23), and (28), we know that the initial value of the closed-loop system Lyapunov function $V\left(x\left(s\right)\right)$ , that is, $V\left(x\left(0\right)\right)$ , is bounded.

Considering $\mathrm{d}V/\mathrm{d}s\le 0$ in (29), we know that $V\left(x\left(s\right)\right)\le V\left(x\left(0\right)\right)$ in $s\in [0,+\infty )$ , that is, $V\left(x\left(s\right)\right)$ is bounded in $s\in [0,+\infty )$ .

Then by using (17) and (23), we know that $\parallel q\left(s\right){\parallel }_2$ , $\parallel z\left(s\right){\parallel }_2$ , and $\parallel \tilde{J}\left(s\right){\parallel }_2$ are bounded in $s\in [0,+\infty )$ , so we obtain the existence and boundedness of $\mathrm{d}V\left(x\left(s\right)\right)/\mathrm{d}s$ from (29). Thus, $V\left(x\left(s\right)\right)$ is uniformly continuous with respect to $s$ in $s\in [0,+\infty )$ .

Considering two constant parameters $b_1\in (0,1)$ , $b_2\in (1,+\infty )$ and two monotonically increasing functions ${\gamma }_1(\parallel x{\parallel }_2)=V\left(b_{1}x\right)$ and ${\gamma }_2(\parallel x{\parallel }_2)=V\left(b_{2}x\right)$ with respect to $\parallel x{\parallel }_2$ , we know that ${\gamma }_1\left(0\right)={\gamma }_2\left(0\right)=0$ , $\mathrm{li}{\mathrm{m}}_{\parallel x{\parallel }_2\to +\infty }{\gamma }_1(\parallel x{\parallel }_2)\to +\infty$ , and $\mathrm{li}{\mathrm{m}}_{\parallel x{\parallel }_2\to +\infty }{\gamma }_2(\parallel x{\parallel }_2)\to +\infty$ , so we know that ${\gamma }_1(\parallel x{\parallel }_2)$ and ${\gamma }_2(\parallel x{\parallel }_2)$ are class ${\mathcal{K}}_{\infty }$ functions and that ${\gamma }_1(\parallel x{\parallel }_2)\le V\left(x\right)\le {\gamma }_2(\parallel x{\parallel }_2)$ holds.

Then we define a scalar function $W\left(x\right)\in \mathbb{R}$ as

$$\begin{array}{c}W\left(x\right)=c_1{q}^{\mathrm{T}}q+c_2{z}^{\mathrm{T}}z.\hfill \end{array}$$

(31)

From equation (29), we know that

$$\frac{\mathrm{d}V\left(x\left(s\right)\right)}{\mathrm{d}s}=-W\left(x\right).$$

(32)

It can be obtained from (32) and Lemma1 that

$$\begin{array}{c}\underset{s\to +\infty }{\lim }W\left(x\left(s\right)\right)=0.\hfill \end{array}$$

(33)

Thus, according to equation (31), we have

$$\begin{array}{c}\underset{s\to +\infty }{\lim }q\left(s\right)=\underset{s\to +\infty }{\lim }z\left(s\right)=0.\hfill \end{array}$$

(34)

Finally, from (11) and (34) we have that

$$\begin{array}{c}\underset{t\to T}{\lim }q\left(t\right)=\underset{t\to T}{\lim }z\left(t\right)=0.\hfill \end{array}$$

(35)

Therefore, according to Lemma 2, it is known that the trajectories of $q\left(t\right)$ and $z\left(t\right)$ are predefined-time stable.

• Step 2. To prove the anti-unwinding property of the attitude control process with contradiction.

According to (34), (35) and the fact $q^{\mathrm{T}}q+q_0^2=1$ , we know that

$$\begin{array}{c}\underset{s\to +\infty }{\lim }{q}_0\left(s\right)=\pm 1,\underset{t\to T}{\lim }{q}_0\left(t\right)=\pm 1.\hfill \end{array}$$

(36)

Considering a case when the initial $q\left(t\right)$ or $q\left(s\right)$ satisfies $q_0\left(0\right)>0$ , we assume that there exists $s_1\in (0,+\infty )$ such that $q_0\left(s_1\right)=-1$ . Since $q\left(s\right)$ is continuous with respect to $s$ in $s\in [0,+\infty )$ , there exists $s_2\in (0,s_1)$ such that $q_0\left(s_2\right)=0$ . Thus, recalling (18) and (23) yields

$$\begin{array}{c}\underset{s\to s_2}{\lim }V\left(x\left(s\right)\right)\to +\infty .\hfill \end{array}$$

(37)

This contradicts the boundedness of $V\left(x\left(s\right)\right)$ in $s\in [0,+\infty )$ . Therefore, for any $s\in (0,+\infty )$ , $q_0\left(s\right)\ne -1$ . Moreover, we know from (11) that for any $t\in [0,T)$ , $q_0\left(t\right)\ne -1$ . Then from (36), we know that

$$\begin{array}{c}\underset{t\to T}{\lim }{q}_0\left(t\right)=1,q_0\left(0\right)>0.\hfill \end{array}$$

(38)

Similarly, it can be obtained that

$$\begin{array}{c}\underset{t\to T}{\lim }{q}_0=-1,q_0\left(0\right)<0.\hfill \end{array}$$

(39)

Therefore, the attitude control process has anti-unwinding property.

• Step 3. To prove the predefined-time stability of $\omega \left(t\right)$ .

From (17), (21), and (34), we know that

$$\begin{array}{c}\multicolumn{1}{c}{\underset{s\to +\infty }{\lim }\omega \left(s\right)=\underset{s\to +\infty }{\lim }z\left(s\right)+\underset{s\to +\infty }{\lim }\alpha \left(s\right)}\\ \multicolumn{1}{c}{=-c_1\underset{s\to +\infty }{\lim }\frac{q\left(s\right)}{K\left(q_0\left(s\right)\right){\mathrm{e}}^{-\frac{s}{T}}}.}\end{array}$$

(40)

Consider a case when $q_0\left(0\right)\ne 0$ . According to (16), (20), (34), (36) and L’Hopital’s rule, we have

$$\begin{array}{c}\multicolumn{1}{c}{\underset{s\to +\infty }{\lim }\omega \left(s\right)=\pm \frac{c_1}{k+1}\underset{s\to +\infty }{\lim }\frac{q\left(s\right)}{{\mathrm{e}}^{-\frac{s}{T}}}}\\ =\pm \frac{c_{1}T}{2(k+1)}\underset{s\to +\infty }{\lim }(q^{\times }\left(s\right)+q_0\left(s\right)I_3)\omega \left(s\right)\hfill \\ =\pm \frac{c_{1}T}{2(k+1)}\underset{s\to +\infty }{\lim }\omega \left(s\right).\hfill \end{array}$$

(41)

Thus, we have that

$$\begin{array}{c}[1\pm \frac{c_{1}T}{2(k+1)}]\underset{s\to +\infty }{\lim }\omega \left(s\right)=0.\hfill \end{array}$$

(42)

Choosing parameters $c_1$ and $k$ such that $c_{1}T\ne 2(k+1)$ yields

$$\begin{array}{c}\underset{s\to +\infty }{\lim }\omega \left(s\right)=0,q\left(0\right)\ne 0.\hfill \end{array}$$

(43)

Considering another case when $q_0\left(0\right)=0$ , we know from (16), (20), (34), (36) and L’Hopital’s rule that

$$\begin{array}{c}\multicolumn{1}{c}{\underset{s\to +\infty }{\lim }\omega \left(s\right)=c_1\underset{s\to +\infty }{\lim }\frac{q\left(s\right)}{{\mathrm{e}}^{-\frac{s}{T}}}}\\ \multicolumn{1}{c}{=\frac{c_{1}T}{2}\underset{s\to +\infty }{\lim }(q^{\times }\left(s\right)+q_0\left(s\right)I_3)\omega \left(s\right)}\\ \multicolumn{1}{c}{=\pm \frac{c_{1}T}{2}\underset{s\to +\infty }{\lim }\omega \left(s\right).}\end{array}$$

(44)

Then we have that

$$\begin{array}{c}(1\pm \frac{c_{1}T}{2})\underset{s\to +\infty }{\lim }\omega \left(s\right)=0.\hfill \end{array}$$

(45)

Choosing parameters $c_1$ and $k$ such that $c_{1}T\ne 2$ yields

$$\begin{array}{c}\underset{s\to +\infty }{\lim }\omega \left(s\right)=0,q\left(0\right)=0.\hfill \end{array}$$

(46)

Thus, combining the results in (43) and (47), we can see that if the parameters $c_1$ and $k$ are chosen such that $c_{1}T\ne 2(k+1)$ and $c_{1}T\ne 2$ , then we can ensure that

$$\begin{array}{c}\underset{s\to +\infty }{\lim }\omega \left(s\right)=0,\underset{t\to T}{\lim }\omega \left(t\right)=0.\hfill \end{array}$$

(47)

Therefore, according to Lemma2, it can be seen that the trajectory of $\omega \left(t\right)$ is predefined-time time stable.

• Step 4. To analyze the performance of the system (1) in $t\ge T$ .

Since $q\left(t\right)$ and $\omega \left(t\right)$ is continuous and differentiable with respect to time for all $t\ge 0$ according to (1), we have from (35) and (47) that $q\left(T\right)=\mathrm{li}{\mathrm{m}}_{t\to T}q\left(t\right)=0$ and $\omega \left(T\right)=\mathrm{li}{\mathrm{m}}_{t\to T}\omega \left(t\right)=0$ . Considering (30), we know that the zero-state with zero-input response of system (1) is given by

$$\begin{array}{c}q\left(t\right)=\omega \left(t\right)=0,t\ge T.\hfill \end{array}$$

(48)

As a result, the states of the spacecraft attitude control system (1), $q\left(t\right)$ and $\omega \left(t\right)$ , are maintained at the equilibrium in $t\ge T$ . This ends the proof.

Finally, in order to facilitate the usage of the control algorithm, we modify (21), (27), and (28) with respect to the transformed time scale $s$ in $t<T$ to the corresponding expressions with respect to the original time scale $t$ according to (9), as

$$\alpha =-\frac{c_{1}T}{T-t}{K}^{-1}\left(q_0\right)q$$

(49)

$$u={\omega }^{\times }L\left(\omega \right)\hat{J}-K\left(q_0\right)q+\frac{-c_2\mathrm{Tz}}{T-t}+L\left(\stackrel{·}{\alpha }\right)\hat{J}$$

(50)

$$\stackrel{·}{\hat{J}}=-\gamma L^{\mathrm{T}}\left(\omega \right){\omega }^{\times T}z-\gamma L^{\mathrm{T}}\left(\stackrel{·}{\alpha }\right)z,t<T$$

(51)

where

$$\stackrel{·}{\alpha }=-\frac{c_{1}T}{T-t}[Y\left(q_0\right){\stackrel{·}{q}}_{0}q+K^{-1}\left(q_0\right)\stackrel{·}{q}+\frac{1}{T-t}{K}^{-1}\left(q_0\right)q].$$

(52)

• Remark 4. The composite barrier Lyapunov function $V_1$ in (18) is selected based on the initial quaternion $q_0\left(0\right)$ . It is not necessary to consider the anti-unwinding issue if $q_0\left(0\right)=0$ according to the control objective (8), so $V_1$ is selected as a quadratic Lyapunov function. However, $V_1$ is chosen as a barrier Lyapunov function if $q_0\left(0\right)\ne 0$ to avoid attitude unwinding. It can be seen that the term $-k\ln {q_0}^2$ in $V_1$ tends to infinity if $q_0\left(t\right)$ crosses over zero. By ensuring the boundedness of $V_1\left(t\right)$ , we can guarantee that $q_0\left(t\right)$ never crosses over zero for anti-unwinding.

• Remark 5. This paper applies LaSalle-Yoshizawa theorem to analyze the system stability, since the closed-loop system Lyapunov function $V\left(x\left(s\right)\right)$ in (23) is positive definite with respect to the closed-loop system error vector $x\left(s\right)=[q^{\mathrm{T}}\left(s\right),z^{\mathrm{T}}\left(s\right),{\tilde{J}}^{\mathrm{T}},\left(s\right){]}^{\mathrm{T}}$ , but its derivative $\mathrm{d}V/\mathrm{d}s$ in (29) is negative definite with respect to the partial error vector $x\left(s\right)=[q^{\mathrm{T}}\left(s\right),z^{\mathrm{T}}\left(s\right){]}^{\mathrm{T}}$ . It can be proven by LaSalle-Yoshizawa theorem that $\mathrm{li}{\mathrm{m}}_{s\to +\infty }q\left(s\right)=\mathrm{li}{\mathrm{m}}_{s\to +\infty }z\left(s\right)=0$ in (34). We can obtain the boundedness of $\parallel \tilde{J}\left(s\right){\parallel }_2$ in $s\in [0,+\infty )$ by considering the boundedness of $V\left(x\left(0\right)\right)$ and the fact $\mathrm{d}V/\mathrm{d}s\le 0$ in $s\in [0,+\infty )$ in (29). Therefore, we know that $\parallel \tilde{J}\left(t\right){\parallel }_2$ is bounded in $t\in [0,T)$ by considering the uniformity between the original time period $t\in [0,T)$ and the transformed time period $s\in [0,+\infty )$ .

• Remark 6. It can be seen in (50) that the predefined settling time $T$ of the spacecraft attitude control system (1) appears directly in the expression of the control law $u$ . It is easy for users to obtain the time and set its value directly.

• Remark 7. This paper proposes a precise predefined-time attitude control method for a spacecraft based on time mapping. Compared with traditional predefined-time attitude control approaches,^30–35 the attitude error after the predefined settling time is exact zero under the proposed method, so control accuracy is significantly improved. Compared with traditional predefined-time attitude control methods,^{28,29,31–33} the proposed method does not assume that the spacecraft inertia information or its basis value is known, so the proposed method is applicable to the spacecraft attitude control system with uncertain inertia after the capturing non-cooperative targets.

Numerical simulation

Simulation of the proposed method with different predefined settling times

In this subsection, numerical simulations are performed to verify the effectiveness of the proposed method. In the simulation, the initial spacecraft attitude quaternion is set as $q_0\left(0\right)=-0.818$ , $q\left(0\right)=[-0.346,-0.175,0.425{]}^{\mathrm{T}}$ , the initial angular velocity is set as $\omega \left(0\right)={[-0.145,0.419,0.374]}^{\mathrm{T}}\mathrm{rad}/\mathrm{s}$ , the parameters in the unknown inertia matrix $J$ are set as $J_{11}=5$ , $J_{12}=0$ , $J_{13}=0$ , $J_{22}=4$ , $J_{23}=0$ , $J_{33}=2$ , and the parameters in the control law are set as $c_1=1$ , $c_2=1$ , $k=1$ , $\gamma =8$ , and the desired predefined settling times $T$ for the spacecraft attitude control process are set as $15\mathrm{s}$ , $25\mathrm{s}$ , and $50\mathrm{s}$ in three simulation cases, respectively. The simulation results are shown in Figures 5 to 8.

Figure 5.

Trajectories of spacecraft attitude quaternion under the proposed controller.

Figure 6.

Trajectories of spacecraft angular velocities under the proposed controller.

Figure 7.

Trajectories of spacecraft control torques under the proposed controller.

Figure 8.

Trajectories of norm of spacecraft inertia estimation under the proposed controller.

It can be seen from Figures 5 and 6 that the trajectories of the spacecraft attitude quaternion $q\left(t\right)$ and angular velocity $\omega \left(t\right)$ converge to zero within the three desired predefined settling times $T$ , respectively. It indicates that the spacecraft attitude control system under the proposed method is capable for accurate predefined-time attitude control with inertia uncertainty. The trajectory of $q_0\left(t\right)$ in Figure 5 does not pass through zero, indicating that the spacecraft attitude control process under the proposed method avoids the unwinding phenomenon. The spacecraft control input signals in Figure 7 are always continuous. The trajectory of the norm of the estimation of unknown inertia information $\parallel \hat{J}\left(t\right){\parallel }_2$ in Figure 8 is bounded. The simulation results confirm the effectiveness of the control method in this paper.

Simulation comparison

This subsection compares the proposed controller with some traditional controllers to verify the superiority of the proposed method. The comparative controllers are selected as the traditional finite-time controller (FTC) in Ref. ¹⁶ and the traditional predefined-time controller (PTC) in Ref. ²⁸ Traditional FTC ¹⁶ assumed that the spacecraft inertia information is unknown, and estimates it with neural networks. Traditional PTC ²⁸ assumed the uncertainty caused by the spacecraft inertia variation as external perturbations, and uses basis inertia information to design the control law.

In the simulation, the initial spacecraft attitude quaternion is chosen as $q_0\left(0\right)=-0.553$ , $q\left(0\right)={[0.613,0.523,-0.212]}^{\mathrm{T}}$ , the initial angular velocity is selected as $\omega \left(0\right)={[-0.613,0.545,-0.736]}^{\mathrm{T}}\mathrm{rad}/\mathrm{s}$ , the parameters in the unknown inertia matrix $J$ are set as $J_{11}=22$ , $J_{12}=0.5$ , $J_{13}=1$ , $J_{22}=19$ , $J_{23}=0.6$ , $J_{33}=16$ . The parameters in the proposed method are set as $c_1=3$ , $c_2=3$ , $k=5$ , and the desired predefined settling time of the spacecraft attitude control process is chosen as $T=30\mathrm{s}$ .

The traditional FTC ¹⁶ is given by:

$$\begin{array}{c}u=N(\varsigma ){\alpha }_1\hfill \\ {\alpha }_1=z_2+\hat{\theta }{s}_2(z_2)+\frac{z_2}{2(k_{b2}^{\mathrm{T}}{k}_{b2}-z_2^{\mathrm{T}}{z}_2)}\hfill \\ \stackrel{·}{\hat{\theta }}=\frac{z_2^{\mathrm{T}}}{k_{b2}^{\mathrm{T}}{k}_{b2}-z_2^{\mathrm{T}}{z}_2}{s}_2(Z_2)-\hat{\theta }\hfill \end{array}.$$

(53)

where $N\left(\varsigma \right)$ is a Nussbaum function selected in Ref., ¹⁶ $z_2=\omega +2q$ , $s_2\left(z_2\right)$ is the radial basis neural network function, $z_2={[q^{\mathrm{T}},{\omega }^{\mathrm{T}}]}^{\mathrm{T}}$ , $\hat{\theta }$ is an estimation of the optimal weight of the neural networks, $k_{b2}={[0.05,0.05,0.05]}^{\mathrm{T}}$ .

The traditional PTC ²⁸ is given by:

$$\begin{array}{c}\multicolumn{1}{c}{u={\omega }^{\times }{J}_0\omega -J_0{\stackrel{·}{\alpha }}_2-\frac{1}{2p_{c2}{T}_{c2}}\exp \left(V_2^{p_2}\right)}\\ \multicolumn{1}{c}{V_2^{p_2}{J}_0\sigma -0.07\mathrm{diag}\left(\frac{{\left({\sigma }^{\mathrm{T}}{J}_0^{-1}\right)}_i}{{\Vert {\sigma }^{\mathrm{T}}{J}_0^{-1}\Vert }_2+\epsilon }\right)\Lambda }\end{array}$$

(54)

where $J_0=\mathrm{diag}\{20,17,15\}$ is the known basis inertia matrix, ${(·)}_i$ denotes the $i$ th element for the vector $(·)\in {\mathbb{R}}^{1\times 3}$ $(i=1,2,3)$ , $\epsilon =0.5$ , $\Lambda \in {\mathbb{R}}^3$ denotes a unit vector, $\sigma =\omega +{\alpha }_2$ , $V_2={\sigma }^{\mathrm{T}}\sigma /2$ , $p_1=p_2=0.16$ , $T_{c1}=T_{c2}=15\mathrm{s}$ . The desired predefined settling time of the spacecraft attitude control process is $T=T_{c1}+T_{c2}=30\mathrm{s}$ . The simulation results are shown in Figures 9 to 11.

Figure 9.

Trajectories of spacecraft attitude quaternion under three controllers.

Figure 10.

Trajectories of spacecraft angular velocity under three controllers.

Figure 11.

Trajectories of spacecraft control torque under three controllers.

It can be seen from Figures 9 and 10 that the trajectories of spacecraft attitude quaternion $q\left(t\right)$ and angular velocity $\omega \left(t\right)$ under the proposed controller and traditional PTC are stabilized within the predefined settling time $T=30\mathrm{s}$ . In Figures 9 and 10, the spacecraft attitude system under traditional FTC is stabilized fast, but it is impossible to know and set the desired settling time boundary with an explicit parameter in the algorithm (53). Moreover, it can be obtained in (53) that the spacecraft attitude control accuracy is reduced due to the introduction of neural network approximation error in traditional FTC, and it can be obtained in (54) that the control accuracy is also reduced due to the introduction of continuous function instead of signum function in traditional PTC in order to avoid control torque chattering. Therefore, Figure 9 shows that the steady state accuracy of spacecraft attitude quaternion under the proposed method is much higher than that under the two comparison controllers. In addition, Figure 9 shows that the unwinding phenomenon occurs in the attitude control process under the traditional FTC. Finally, the spacecraft control torque trajectories under the three methods are bounded in Figure 11. But the absolute value of peak value of the control torques under the proposed controller is not more than $8.30\mathrm{N}·\mathrm{m}$ , that is less than $35.23\mathrm{N}·\mathrm{m}$ under traditional PTC and $12.49\mathrm{N}·\mathrm{m}$ under traditional FTC. As a conclusion, the proposed controller is applicable to the spacecraft attitude control system with unknown inertia information, and it has the comprehensive advantages of anti-unwinding characteristic, accurate control and direct settling time adjustment, compared with the traditional FTC ¹⁶ and the traditional PTC. ²⁸

Conclusion and future work

This paper proposes a new adaptive predefined-time precise anti-unwinding attitude controller for a spacecraft with inertia uncertainty. A time-mapping function containing the predefined settling time is introduced to transform the time scale of the original attitude control system. A composite barrier Lyapunov function is utilized to avoid the unwinding phenomenon in the attitude control process. It is proved that the settling time of attitude control system can be specified as an explicit parameter in the control algorithm, and the attitude error after the predefined settling time is exact zero. Simulation examples show that the presented controller accomplishes the accurate predefined-time attitude control of a spacecraft with unknown inertia. Comparative simulations with the traditional methods indicate that the proposed controller has the advantages of anti-unwinding attitude control and improved steady attitude accuracy.

In future work, adaptive predefined-time precise attitude formulation control for a multi-spacecraft system with inertia uncertainties can be studied.